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A new description of space and time using Clifford multivectors

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A new description of space and time using Clifford multivectors
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    a  r   X   i  v  :   1   2   0   5 .   5   1   9   5  v   2   [  m  a   t   h  -  p   h   ]   1   1   O  c   t   2   0   1   2 A new description of space and time using Cliffordmultivectors James M. Chappell † , Nicolangelo Iannella † , Azhar Iqbal † , Mark Chappell ‡ , DerekAbbott † † School of Electrical and Electronic Engineering, University of Adelaide, South Australia 5005,Australia  ‡ Griffith Institute, Griffith University, Queensland 4122, Australia  Abstract Following the development of the special theory of relativity in 1905, Minkowski pro-posed a unified space and time structure consisting of three space dimensions and onetime dimension, with relativistic effects then being natural consequences of this space-time geometry. In this paper, we illustrate how Clifford’s geometric algebra that utilizesmultivectors to represent spacetime, provides an elegant mathematical framework for thestudy of relativistic phenomena. We show, with several examples, how the application of geometric algebra leads to the correct relativistic description of the physical phenomenabeing considered. This approach not only provides a compact mathematical representa-tion to tackle such phenomena, but also suggests some novel insights into the nature of time. Keywords:  Geometric algebra, Clifford space, Spacetime, Multivectors, Algebraicframework 1. Introduction The physical world, based on early investigations, was deemed to possess three inde-pendent freedoms of translation, referred to as the three dimensions of space. This naiveconclusion is also supported by more sophisticated analysis such as the existence of onlyfive regular polyhedra and the inverse square force laws. If we lived in a world with fourspatial dimensions, for example, we would be able to construct six regular solids, and infive dimensions and above we would find only three [1]. Gravity and the electromagneticforce laws have also been experimentally verified to follow an inverse square law to veryhigh precision [2], indicating the absence of additional macroscopic dimensions beyondthree space dimensions. Additionally Ehrenfest [3] showed that planetary orbits are notstable for more than three space dimensions, with Tangherlini [4] extending this resultfor electronic orbitals around atoms. The importance of the inverse square force law canbe seen from Bertrand’s theorem from classical mechanics, which states that ‘The only Email address:  james.m.chappell@adelaide.edu.au  (James M. Chappell † ) Preprint submitted to Physics Letters A October 12, 2012   central forces that result in closed orbits for all bound particles are the inverse-squarelaw and Hooke’s law.’ [5].The concept of time is typically modeled as a fourth Euclidean-type dimension ap-pended to the three dimensions of spatial movement forming Minkowski spacetime [6]with spacetime events modeled as four-vectors. Alternative algebraic formalisms thoughhave been proposed such as STA (Space Time Algebra) [7, 8, 9, 10, 11], or APS (Algebraof Physical Space) [12]. The representation of time within the Minkowski framework andalso these alternate approaches interpret time as a Cartesian-type dimension. Howeverwith the observed non-Cartesian like behavior of time, such as the time axis possessinga negative contribution to the Pythagorean distance, and the observed inability to freelymove within the time dimension as is possible with space dimensions suggests that analternate representation might be preferable. The results of string theory also suggestthat the ordinary formulation of physics, in a space-time with three space dimensionsand one time dimension, is insufficient to describe our world [13, 14, 15]. Also, Tifftexplained the apparent quantization of the cosmological redshift [16, 17] with a modelusing time with three dimensions [18, 19, 20, 21], additionally the spinorial nature of time noted by another author [22].Chappell et al. [23] recently employed a two-dimensional multivector as an algebraicmodel for spacetime in the plane, with time represented as a bivector. A bivector ingeometric algebra represents an oriented unit area, which acts as a rotation operatorfor the plane. Now, through generalizing this two-dimensional multivector descriptionto three-dimensions we produce a three dimensional vector  x  and time becomes a unitarea oriented within three dimensions, represented as  ic t , which can be combined intoa multivector  x  +  ic t . This now gives rise to symmetry between the space and timecoordinates, both being represented as three-vectors in our framework, while still pro-ducing the results of special relativity. In fact this duality between space and time, withboth possessing the same degrees of freedom, is only possible in three dimensions—intwo dimensions for example we have two degrees of freedom for translation but only onefor rotation [24, 25].Clifford’s geometric algebra was first published in 1873, extending the work of Grass-man and Hamilton, creating a single unified real mathematical framework over Carte-sian coordinates, which naturally included the algebraic properties of scalars, complexnumbers, quaternions and vectors into a single entity, called the multivector [26]. Wefind that this general algebraic entity, as part of a real three-dimensional Clifford algebra Cl  3 , 0 ( ℜ ), provides an equivalent representation to a Minkowski vector space ℜ 3 , 1 [27, 28],with significant benefits in assisting intuition and in providing an efficient representation.In order to represent the three independent dimensions of space, Clifford defined threealgebraic elements  e 1 ,  e 2  and  e 3 , with the expected unit vector property e 21  =  e 22  =  e 23  = 1 (1)but with each element anticommuting, that is  e i e j  = − e j e i , for  i  =  j . We then find thatthe composite algebraic element, the trivector  i  =  e 1 e 2 e 3  squares to minus one, that is, i 2 = ( e 1 e 2 e 3 ) 2 =  e 1 e 2 e 3 e 1 e 2 e 3  =  − e 1 e 1 e 2 e 2  =  − 1, assuming an associative algebra, andas it is found to commute with all other elements it can be used interchangeably with theunit imaginary i = √ − 1. We continue to follow this notational convention throughoutthe paper, denoting the Clifford trivector with  i , the scalar imaginary i = √ − 1 and the2  bivector of the plane with an iota,  ι  =  e 1 e 2 . A general Clifford multivector for three-spacecan be written by combining all available algebraic elements a  + x 1 e 1  + x 2 e 2  + x 3 e 3  + i ( t 1 e 1  + t 2 e 2  + t 3 e 3 ) + ib,  (2)where  a ,  b ,  x k  and  t k  are real scalars, and  i  is the trivector. We thus have eight degreesof freedom present, in which we use  x  =  x 1 e 1 + x 2 e 2 + x 3 e 3  to represent a Cartesian-typevector, and the bivector  i ( t 1 e 1  +  t 2 e 2  +  t 3 e 3 ) =  t 1 e 2 e 3  +  t 2 e 3 e 1  +  t 3 e 1 e 2  to representtime, forming a time vector  t  =  t 1 e 1  + t 2 e 2  + t 3 e 3 . Thus in our framework the temporalcomponents are described by the three bivector components of the multivector. However,when we move into the rest frame of the particle, we only require a single bivectorcomponent, so that time becomes one dimensional as conventionally assumed. That is  ℜ 3 is the exterior algebra of   ℜ 3 which produces the space of multivectors  ℜ⊕ℜ 3 ⊕  2 ℜ 3 ⊕   3 ℜ 3 , an eight-dimensional real vector space denoted by  Cl 3 , 0 ( ℜ ), with thecomplex numbers and quaternions as subalgebras. The presence of the quaternions as asub-algebra, represented as  a  +  i t , is significant as Hamilton showed that they providethe correct algebra for rotations in three dimensions. The complex numbers using thedescription in Eq. (2) would be described as  a + ib  or by using one of the bivectors, suchas  a  + ie k .It might be argued that four dimensional spacetime is required to describe the Diracequation, for example, typically described using the four dimensional Dirac matrices as γ  µ ∂  µ | ψ   =  − i m | ψ  , where i = √ − 1 and  | ψ   is an eight dimensional spinor. Howeverit has been shown [29, 30] that it can be reduced to an equivalent equation in threedimensional space using  Cl 3 , 0 ( ℜ ), being written as ∂ψ  = − mψ ∗ ie 3 ,  (3)with the multivector gradient operator  ∂   =  ∂  t + ∇ , with ∇ being the three-gradient, and ψ  the multivector of three dimensions, shown in Eq. (2), and the operation  ψ ∗ flippingthe sign of the vector and trivector components. This can also be compared with thewell known form of Maxwell’s equations in three-space  ∂ψ  =  J  , with the electromagneticfield represented by the multivector  ψ  =  E  +  i B  and sources  J   =  ρ  −  J . Hence themultivector of   Cl 3 , 0 ( ℜ ) unifies complex numbers, spinors, four-vectors and tensors intoa single algebraic object. We now utilize the basic properties of the Clifford multivectorto produce an alternate algebraic representation of space and time. Clifford multivector spacetime It was shown previously [23] that spacetime events can be described with the multi-vector of two dimensions X   =  x + ιct,  (4)having the appropriate properties to reproduce the results of special relativity, assuminginteractions are planar, with  x  =  x 1 e 1  +  x 2 e 2  representing a vector in the plane and t  the observer time, and  ι  =  e 1 e 2  the bivector of the plane. Squaring the coordinatemultivector we find  X  2 =  x 2 − c 2 t 2 , thus producing the correct spacetime distance. Thisrepresentation of coordinates, along with an electromagnetic field  F   =  E + ιcB , are sub- ject to a Lorentz transformation of the form  L  = e φ ˆ v / 2 e ιθ/ 2 , with the first term defining3  boosts and the second term rotations, where the transformed coordinates or transformedfield are given by  X  ′ =  LXL † , where  L † = e − ιθ/ 2 e − φ ˆ v / 2 . That is, a combined boost anda rotation can be written as X  ′ = e φ ˆ v / 2 e ιθ/ 2 X  e − ιθ/ 2 e − φ ˆ v / 2 .  (5)We can see that the multivector in Eq. (4) essentially sets up two orthogonal directions  x and  ι  =  ie 3 , and so we can generalize this structure to three dimensions through a generalrotation of the  e 1 e 2  plane into three-space forming the three-dimensional multivector X   =  x + ic t ,  (6)where  x  =  x 1 e 1  +  x 2 e 2  +  x 3 e 3  is the coordinate vector and  t  =  t 1 e 1  +  t 2 e 2  +  t 3 e 3  atime vector, with  i  =  e 1 e 2 e 3  the trivector. To maintain the orthogonality implied by themultivector in two dimensions we therefore have the constraints  x · t  = 0. Now, squaringthe multivector we find X  2 = ( x + ic t )( x + ic t ) (7)=  x 2 − c 2 t 2 + 2 ic x · t  =  x 2 − c 2 t 2 using the fact that  xt  +  tx  = 2 x · t  = 0, referring to Eq. (A.3) in the Appendix, thusproducing the correct spacetime distance in three dimensions. Thus an event can bedescribed by a position vector in three dimensions, and a three dimensional time vectorin our framework, and to maintain Lorentz invariance we require these two vectors tobe orthogonal. Hence Einstein’s assertion of the invariant distance in Eq. (7) in histheory of special relativity, from this new viewpoint, becomes simply an assertion of theorthogonality of the space and time three-vectors describing an event in Eq. (6).We have from Eq. (4) the multivector differential dX   =  d x + icd t .  (8)For the rest frame of the particle we have  dX  20  =  − c 2 dτ  2 , which defines the propertime  τ   of the particle. In the co-moving frame of the particle the time vector does notrequire three components with which to describe its orientation, but only has one degreeof freedom defining its length. We have assumed that the speed of light  c  is the samein the rest and the moving frame, as required by Einstein’s second postulate. Now, if the spacetime distance defined in Eq. (7) is invariant under the Lorentz transformationsdefined later in Eq. (16), then we can equate the rest frame interval to the moving frameinterval, giving c 2 dτ  2 =  c 2 d t 2 − d x 2 =  c 2 d t 2 − v 2 d t 2 =  c 2 d t 2  1 −  v 2 c 2  ,  (9)assuming  d x 2 =  v 2 d t 2 , a vector equation that we confirm shortly. Taking the squareroot of Eq. (9), we find the time dilation formula  | d t | =  γdτ   where γ   = 1   1 − v 2 /c 2 .  (10)4  This equation implies that the length of the time vector  | d t | , in this formalism, equatesto the scalar time variable  t  typically employed in special relativity. From Eq. (8), wecan now calculate the proper velocity through differentiating with respect to the scalarproper time, giving a multivector representing velocity U   =  dX dτ   =  d x | d t || d t | dτ   + icd t dτ   =  γ  ( v  + i c ) ,  (11)where we use  | d t | dτ   =  γ   and  v  =  d x | d t |  and the speed of light now becomes vectorial in thedirection of the time vector  c  =  c ˆ t . We can then find U  2 =  γ  2 ( v + i c ) 2 =  1 −  v 2 c 2  − 1 ( v 2 − c 2 + 2 i v · c ) = − c 2 ,  (12)with the orthogonality of space and time  x · t  = 0, implying  v · c  = 0. We define themomentum multivector P   =  mU   =  γm v + iγm c  =  p + i E c ,  (13)with the relativistic momentum  p  =  γm v  and the total energy now having a vectorialnature  E  =  γmc c .Now, as  U  2 =  − c 2 , then  P  2 =  − m 2 c 2 is an invariant describing the conservation of momentum and energy, which gives P  2 c 2 =  p 2 c 2 − E 2 = − m 2 c 4 ,  (14)or  E 2 =  m 2 c 4 +  p 2 c 2 , the relativistic expression for the conservation of momentum-energy, using the orthogonality of momentum and energy from  p · E  =  γ  2 m 2 c ( v · c ) = 0.The vectorial nature of energy, analogous to the Poynting vector describing energy flow,is possible because for a Clifford vector  E , we find that  E 2 is in general a scalar quantityand so can satisfy the Einstein energy-momentum relation as shown. At rest the energy E 0  =  mc c  has a vectorial nature and hence the Einstein momentum-energy relationin Eq. (14) can be naturally interpreted as a Pythagorean triangle relation between aconstant rest energy vector  mc c  and the momentum vector  p c . 1.1. The Lorentz Group The Lorentz transformations describe the transformations for observations betweeninertial systems in relative motion. The set of transformations describing rotationsand boosts connected with the identity is referred to as the restricted Lorentz group SO + (3 , 1). We find that the exponential of the bivector e i ˆ u θ , describes rotations in theplane  i ˆ u , as shown in Eq. (B.3), however, more generally, we can define the exponentialof a full multivector  M   defined as in Eq. (2), by constructing the Taylor seriese M  = 1 + M   +  M  2 2! +  M  3 3! + ...,  (15)which is absolutely convergent for all multivectors  M   [31]. We find that the exponentialof a pure vector e φ ˆ v describes boosts, and so if we define the combined operator consistingof a boost and rotation L  = e φ ˆ v e i ˆ w θ ,  (16)5
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